Zero-Divisor Graphs and Lattices of Finite Commutative Rings

Rose-Hulman Undergraduate Mathematics Journal

Volume 12 Issue 1 Article 4

Darrin Weber Millikin University, [email protected]

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Recommended Citation Weber, Darrin (2011) "Zero-Divisor Graphs and Lattices of Finite Commutative Rings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 12 : Iss. 1 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol12/iss1/4 Rose- Hulman Undergraduate Mathematics Journal

Zero-Divisor Graphs and Lattices of Finite Commutative Rings

Darrin Webera

Volume 12, no. 1, Spring 2011

Sponsored by

Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Millikin University, Decatur, IL, [email protected] Rose-Hulman Undergraduate Mathematics Journal Volume 12, no. 1, Spring 2011

Zero-Divisor Graphs and Lattices of Finite Commutative Rings

Darrin Weber

Abstract.In this paper we consider, for a finite commutative ring R, the well- studied zero-divisor graph Γ(R) and the compressed zero-divisor graph Γc(R) of R and a newly-defined graphical structure — the zero-divisor lattice Λ(R) of R. We give results which provide information when Γ(R) =∼ Γ(S), Γc(R) =∼ Γc(S), and Λ(R) =∼ Λ(S) for two finite commutative rings R and S. We also provide a theorem which says that Λ(R) is almost always connected.

Acknowledgements: This paper is the result of a summer’s worth of undergraduate research at Millikin University and was funded by Millikin University’s Student Undergrad- uate Research Fund. The author would like to thank Dr. Joe Stickles for his guidance and advice throughout the development of this paper, Dr. Michael Axtell for his comments and suggestions, and Dr. James Rauﬀ for his helpful input. Page58 RHITUndergrad.Math.J.,Vol. 12, no. 1

1 Introduction

The notion of a zero-divisor graph was introduced in [7] and was studied further in [4]. However, the definition we will provide of the zero-divisor graph of a ring is more generally accepted and was first used in [3]. Much research has been done on zero-divisor graphs over the past ten years, and many of the papers can be found in the reference section of [2]. The hope is that the graph-theoretic properties of the zero-divisior graph will help us better understand the ring-theoretic properties of R. Not only have zero-divisor graphs been studied by professional mathematicians, but they have also been the focus of master’s theses and doctorial dissertations. Further, many under- graduates have researched zero-divisor graphs extensively providing a number of substantial results in the field. In particular, the Wabash Summer Institute in Mathematics at Wabash College has, in part, focused on the interplay between ring structure and zero-divisor graph structure. Some tools used to aid in the research process in this program were Mathematica notebooks that displayed the zero-divisor graph of certain rings. These notebooks have been rewritten as well as additional notebooks added, and they can be found at [14]. All of the graphs displayed in this paper were generated using those notebooks. To help study large zero-divisor graphs, we introduce another related definition to the zero-divisor graph, called the compressed zero-divisor graph, which first appeared in a similar form in [12] and was further studied in [13]. We also introduce the definition for the zero- divisor lattice suggested by Dr. Nicholas Baeth of the University of Central Missouri. In section 2, we provide the necessary definitions for this paper. In section 3, we look at the connections between the zero-divisor graph and the compressed zero-divisor graph and prove that isomorphic graphs yield isomorphic compressed graphs in Theorem 3.1. In Theorem 3.3, we also demonstrate that in most cases cut-sets are preserved when looking at the compressed zero-divisor graph. In section 4, we explore the connections between all three graphical structures and show in Theorems 4.1 and 4.2 that isomorphic graphs or isomorphic compressed graphs gives us isomorphic lattices. In section 5, we show that the zero-divisor lattice is connected in almost every case.

2 Deﬁnitions

Throughout this paper, R denotes a finite commutative ring with identity. An element a is a zero-divisor if there exists a nonzero r ∈ R such that ar = 0. We denote the set of all zero- divisors in R as Z(R). The set of annihilators of a ring element x is ann(x)= {a | ax =0}. A ring is called local if it has one unique maximal ideal. (A maximal ideal is an ideal A of a ring R such that if A ⊆ B ⊆ R, where B is also an ideal, then either A = B or B = R.) A field is a commutative ring with identity in which every nonzero element has a multiplicative inverse. For a graph G, we denote the set of vertices of G as V (G) and the set of edges as E(G). We define a path between two elements a1,an ∈ V (G) to be an ordered sequence of distinct vertices and edges {a1,e1,a2,...,en−1,an} of G such that edge ei, denoted by ai RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 59

— ai+1, is incident to vertices ai and ai+1, for each i ∈ {1,...,n − 1}. For x,y ∈ V (G), the minimum length of all paths from x to y, if it exists, is called the distance from x to y and is denoted d(x,y). If no path from x to y exists, then d(x,y) = ∞. The diameter of a graph is diam(G) = sup{d(x,y) | x,y ∈ V (G)}. The neighborhood of a vertex is the set nbd(x) = {z ∈ V (G) | x — z}. A graph is connected if a path exists between any two distinct vertices. A complete r-partite graph is the disjoint union of r nonempty vertex sets in which two distinct vertices are adjacent if and only if they are in distinct vertex sets. In the case where r = 2, we call the graph complete bipartite. Two graphs G and H are said to be isomorphic if there exists a one-to-one and onto function φ : V (G) → V (H) such that if x,y ∈ V (G), then x — y if and only if φ(x)— φ(y). Lemma 2.1 shows that graph isomorphism preserves neighborhoods. We provide a proof for completeness.

Lemma 2.1. Let G ∼= H. If φ(x)= y, then φ(nbd(x)) = nbd(y). Proof. Let φ : G → H be a graph isomorphism. Let x ∈ V (G) and φ(x)= y ∈ V (H). Then φ(nbd(x)) = {φ(z) | x — z} = {φ(z) | φ(x) — φ(z)} = {φ(z) | y — φ(z)} = nbd(y). The zero-divisor graph of a ring R, denoted Γ(R), is a graph with V (Γ(R)) = Z(R)\{0} and E(Γ(R)) = {a — b | ab = 0}. By [3], we know that Γ(R) is always connected and diam(Γ(R)) ≤ 3 for any ring R. Notice that nbd(x) = ann(x) in a zero-divisor graph. A cut-set of a graph G is a set A ⊂ V (G) minimal among all subsets of V (G) such that there exist distinct vertices c,d ∈ V (G)\A such that every path from c to d involves at least one element of A.

Example 2.2. In Γ(Z30), shown in Figure 1(a), there are three cut-sets: {15}, {10, 20}, and {6, 12, 18, 24}. The cut-sets in Γc(R) are {15}, {10}, and {6} and are shown in Figure 1(b). A cut vertex is a cut-set of size 1. The study of cut vertices of zero-divisor graphs began in [6] and was continued in [9]. In [6], it was shown that a cut vertex along with zero form an ideal in the ring. In [9], the cut vertex was generalized to cut-sets, and cut-sets were classified for finite, nonlocal commutative rings. In addition, it was shown that the cut-set along with zero form an ideal in the ring. For algebraic definitions and concepts not listed here, see [10], and for graph theory definitions and concepts, see [8]. To define both the compressed zero-divisor graph and the zero-divisor lattice, we first need to define an equivalence relation on the zero-divisors of R. Definition 2.1. Let R be a commutative ring. Define a relation ≡ on R by x ≡ y if and only if ann(x) = ann(y). It is easy to see that ≡ is an equivalence relation on R. We denote the equivalence class of x byx ¯. Notice that ann(0) = R andx ¯ =y ¯ for all x,y ∈ R\Z(R). However, we will only focus on the equivalence classes of the nonzero zero-divisors. Page60 RHITUndergrad.Math.J.,Vol. 12, no. 1

Deﬁnition 2.2. For a ring R, the compressed zero-divisor graph, denoted Γc(R), is a graph whose vertices are the equivalence classes of the nonzero zero-divisors, and two vertices a¯ and ¯b are connected by an edge if and only if ab =0.

Example 2.3. For Z30, Figure 1 shows the diﬀerence between the zero-divisor graph, (a), and the compressed zero-divisor graph, (b).

2 3

2 3 4 9 8 15 10 21 10 14 20 27 15

16 6 6 12 22 18 24 26 28 5

25 5

(a) Γ(Z30) (b) Γc(Z30)

Figure 1: The zero-divisor graph and compressed zero-divisor graph of Z30

Notice that by Theorem 2.8 in [3], when the zero-divisor graph is a complete graph, either the ring is Z2 × Z2, or every vertex loops to itself. In the case that R ∼= Z2 × Z2, Γ(R) ∼= Γc(R). Every other complete zero-divisor graph compresses into a single vertex in the compressed zero-divisor graph. To get the definition of a zero-divisor lattice, note that we can put a partial order on V (Γc(R)) by definingx< ¯ y¯ if ann(x) ann(y). Definition 2.3. For a ring R, the zero-divisor lattice, denoted Λ(R), is a lattice where the vertices are the equivalence classes of V (Γ(R)) and there is an edge y¯ → x¯ if and only if x<¯ y¯ and there does not exist z¯ with x<¯ z¯ < y¯.

Example 2.4. Figure 2 displays the zero-divisor lattice of Z30. A zero-divisor lattice is said to be connected if, when the edges are considered to be undirected, you can reach any vertex y from any other vertex x.A root of the zero-divisor lattice is a vertex x such that for every other vertex y, either y

15 10 6

5 3 2

Figure 2: Λ(Z30)

Theorem 2.5. [11, Theorem 80] Let R be a Noetherian ring, and let A be a ﬁnitely generated non-zero R-module. Then there are only a ﬁnite number of maximal primes of A, and each is the annihilator of a non-zero element of A.

Corollary 2.6. Let R be a ﬁnite commutative ring with identity. Then the maximal ideals of R can be realized as the annihilator sets of single elements.

Note that this theorem only applies to the maximal ideals of the ring. For example, in 2 the ring Z4[x]/(x +2x), the set {0, 2x} forms an ideal in the ring, but this ideal cannot be realized by the annihilator of a single element. The following lemma is well-known and will be used in Section 5.

Lemma 2.7. If R ∼= R1 ×× Ri ×× Rn, then the maximal ideals of R have the form M = R1 ×× Mi ×× Rn where Mi is a maximal ideal in Ri.

The next lemma is also a well-known result, and we provide a proof for completeness.

Lemma 2.8. Let R ∼= R1 ×× Ri ×× Rn and let M = R1 ×× Mi ×× Rn be a maximal ideal. Then M = ann((0, 0,...,li,..., 0)) where Mi = ann(li) for some li ∈ Ri.

Proof. It suﬃces to show M = ann((0,...,li,..., 0)). Let (r1,...,rn) ∈ M. Then ri ∈ ann(li), so(r1,...,rn)(0,...,li,..., 0) = (0, 0,..., 0). Hence (r1,...,rn) ∈ ann((0,...,li,..., 0)). Let (r1,...,rn) ∈ ann((0,...,li,..., 0)). Then (r1,...,rn)(0, ...,li,..., 0) = (0, 0,..., 0), which implies rili = 0. Thus, ri ∈ ann(li), and (r1,...,rn) ∈ M.

3 Connections Between Γ(R) and Γc(R)

To start, we show that the compressed zero-divisor graph preserves certain properties of the zero-divisor graph.

Theorem 3.1. Let R and S be ﬁnite commutative rings. If Γ(R) ∼= Γ(S), then Γc(R) ∼= Γc(S). Page62 RHITUndergrad.Math.J.,Vol. 12, no. 1

Proof. Suppose V (Γ(R)) = {r1,r2,...,rn} and V (Γ(S)) = {s1,s2,...,sn} such that the isomorphism φ : Γ(R) → Γ(S) satisﬁes φ(ri)= si for each i ∈ {1, 2,...,n}. By Lemma 2.1, φ(ann(ri)) = ann(si) for each i, and the mapping of edges φ : E(Γc(R)) → E(Γc(S)) which sends the edger ¯i —r ¯j in Γ(R) to the edges ¯i —s ¯j in Γ(S) is a well-deﬁned bijection. Thus, Γc(R) ∼= Γc(S).

The converse of this theorem is false as illlustrated in Example 3.2. Take Z2p and Z2q, where p,q are distinct primes. We have that Γc(Z2p) ∼= Γc(Z2q) but Γ(Z2p) ≇ Γ(Z2q). Example 3.2. Figure 3 displays the zero-divisor graphs and compressed zero-divisor graphs of Z10 and Z14.

5 2 7 2

(a) Γc(Z10) (b) Γc(Z14)

2 8 2 12

5 4 7 10

4 6 6 8

Figure 3: The zero-divisor graphs and compressed zero-divisor graphs of Z10 and Z14

The next result has been proven in more generality in [5]. We provide an alternate proof for the ﬁnite case.

Theorem 3.3. Let R be a ﬁnite commutative ring such that Γ(R) is not complete r-partite. A set A is a cut-set in Γ(R) if and only if A¯ is a cut-set in Γc(R).

Proof. For ease of notation, let the set of vertices A = {a1,a2,...,an} and let A¯ = {a¯1, a¯2,..., a¯n}, where A¯ is the of equivalence classes for all elements in A. (⇒) Let Γ(R) be a zero-divisor graph that is not complete r-partite and let A be a cut-set in Γ(R). Since Γ(R) is connected [3, Theorem 2.3], we have A = ∅. Then there exists distinct c,d ∈ V (Γ(R))\A such that every path from c to d involves at least one vertex in A. Case 1: Assume for all c,d ∈ V (Γ(R))\A, we have c ≡ d. Since Γ(R) is not complete r-partite, then there must exist ai,aj ∈ A such that aiaj = 0. Notice that diam(Γ(R)) ≤ 3 by [3, Theorem 2.3] and that diam(Γ(R)) = 3 since all c,d are connected to every element in the cut-set A. So by Theorem 4.5 in [6], Γ(R) is star-shaped reducible. By Theorem 2.3 in [6], Z(R) forms an ideal. Consider c + ai. If c + ai ∈ A, then (c + ai)d = 0. However, RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 63

(c + ai)d = cd = 0, since c and d are separated by A. So, c + ai ∈ V (Γ(R))\A which means that (c + ai)A = 0, however, (c + ai)aj = aiaj = 0. Hence, c + ai ∈/ V (Γ(R))\A and therefore c + ai ∈/ Z(R). Thus, this case is not realizable as a zero-divisor graph. Case 2: There exists c,d ∈ V (Γ(R))\A such that c ≡ d. Thenc, ¯ d¯ also exists as distinct vertices in Γc(R). If ai ≡ aj for some i = j and 1 ≤ i, j ≤ n, then leta ¯i represent the equivalence class of ai in the graph. If the path from c to d involved an element in {a¯i} in Γ(R), then the path goes througha ¯i in Γc(R). So, A¯ = {a1,..., a¯i,...,an} separates Γc(R). If A¯ was not minimal, then there would exist a f,¯ g¯ ∈ V (Γc(R))\A¯ such that A¯\{a¯i} would separate f¯ andg ¯. However, this would mean that f and g would be separated by A\{a¯i}, which is a contradiction on the minimality of A. Thus, A¯ is a cut-set in Γc(R).

If ai ≡ aj for all i = j and 1 ≤ i, j ≤ n, then any path betweenc ¯ and d¯ still involves at least one vertex in A¯. So, A¯ separates the graph in Γc(R). If A¯ is not minimal in Γc(R), then there exists ana ¯i ∈ A¯ andc, ¯ d¯ ∈ V (Γc(R))\A¯ such thatc ¯ and d¯ are separated by A¯\{a¯i}. This would mean that c and d are separated by A\{a¯i} in Γ(R), which is a contradiction on the minimality of A. Thus, A¯ is minimal and is therefore a cut-set in Γc(R).

(⇐) Let A¯ be a cut-set in Γc(R). Then there exist distinctc, ¯ d¯∈ V (Γc(R))\A¯ such that every path involves at least one element of A¯. Notice that A separates Γ(R) because if it did not, then there would existc, ¯ d¯ that does not involve A¯.

Case 1: Let A¯ = {a¯i} for some ai ∈ A. Every path from c to d passes through A and for all aj ∈ {a¯i}, there exists a path from c to d that involves aj. Since every ai ≡ aj for all ai,aj ∈ A, if there is an edge x — ai for some x ∈ V (Γ(R)), then x is connected to ai for all i. Thus, A is minimal in Γ(R) and is therefore a cut-set.

Case 2: There exist distinct ai,aj ∈ A such that ai ≡ aj. If A is not a cut-set, then there exists an ak ∈ A such that A\{ak} separates Γ(R). Thus, for all c,d ∈ V (Γ(R))\A there exists a path that does not involve ak. Therefore, there exists a path fromc ¯ to d¯ in Γc(R) that does not involvea ¯k. Thus, A¯\{a¯k} separates Γc(R), which is a contradiction on the minimality of A¯. Hence, A is minimal and therefore is a cut-set in Γ(R).

Recall that a complete r-partite zero-divisor graph is the disjoint union of r nonempty vertex sets and two distinct vertices are adjacent if and only if they are in distinct vertex sets. By Theorem 3.1 in [1], if Γ(R) is complete r-partite then |Vi| > 1 for at most one 2 1 ≤ i ≤ r, and if Vj = {x} then x = 0. This means that for all x,xk in the vertex sets of order 1, ann(x) = ann(xk), which means that they are in the same equivalence class and will appear as a single vertex in the compressed zero-divisor graph. Also, for all vertices b,bm ∈ Vi such that |Vi| > 1, ann(b) = ann(bm), which means that they are all in the same equivalence class. Thus, every complete r-partite graph compresses into a graph with two vertices that are connected to each other. Since there are only two vertices, there can be no cut-set. Page64 RHITUndergrad.Math.J.,Vol. 12, no. 1

4 Connections between Γc(R) and Λ(R) In [9], cut-sets were classiﬁed for all ﬁnite, nonlocal rings as annihilator ideals. Notice that 2 in the local ring Z8[x]/(x +2x), shown in Figure 4, a cut-set is {2x, 4x, 2x +4} in the compressed zero-divisor graph. By Theorem 3.3, we know that this corresponds to a cut-set in the zero-divisor graph, which is {2x, 4x, 6x, 2x+4, 6x+4}. Notice that this cut-set (union 2 with {0}) is not an ideal in the ring. Also, in the ring Z4[x]/(x +2x), the vertex 2x is a cut vertex in the zero-divisor graph, but {0, 2x} cannot be realized as the annihilator of a 2 single element. However, we can identify the cut-set of 2x as the root in Λ(Z4[x]/(x +2x)). Because of both of these examples, we hope that studying zero-divisor lattices will help us understand more about the structure and properties of cut-sets.

2 x + 4 4

x + 4 2

2 x 4 x x

3 x + 2 x + 2

2 Figure 4: Γc(Z8[x]/(x +2x))

We begin by proving two theorems relating the structure of Γ(R), Γc(R), and Λ(R). Theorem 4.1. Let R and S be ﬁnite commutative rings. If Γ(R) ∼= Γ(S), then Λ(R) ∼= Λ(S).

Proof. Suppose V (Γ(R)) = {r1,r2,...,rn} and V (Γ(S)) = {s1,s2, ...,sn} such that the isomorphism φ : Γ(R) → Γ(S) satisﬁes φ(ri) = si for each i ∈ {1, 2,...,n}. By Lemma 2.1, φ(ann(ri)) = ann(si) for each i, and if ann(ri) = ann(rj) for any 1 ≤ i, j ≤ n, then ann(si) = ann(sj), and if ann(ri) ⊆ ann(rj) for any i = j, then ann(si) ⊆ ann(sj). Thus, the mapping of edges φ : E(Λ(R)) → E(Λ(S)) which sends the edger ¯i −→ r¯j in Λ(R) to the edges ¯i −→ s¯j in Λ(S) is a well-deﬁned bijection. Thus, Λ(R) ∼= Λ(S). The converse of this theorem is false for the same reason that the converse for Theorem 3.1 is false. An example is given in Figure 5.

Theorem 4.2. Let R and S be ﬁnite commutative rings. If Γc(R) ∼= Γc(S), then Λ(R) ∼= Λ(S).

Proof. Suppose V (Γc(R)) = {r1,r2,...,rn} and V (Γc(S)) = {s1,s2, ...,sn} such that the isomorphism φ : Γc(R) → Γc(S) satisﬁes φ(ri) = si for each i ∈ {1, 2,...,n}. By Lemma 2.1, φ(ann(ri)) = ann(si) for each i, and if ann(ri) = ann(rj) for any 1 ≤ i, j ≤ n, then RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 65

4 9

2 3

(a) Λ(Z8) (b) Λ(Z27)

3 6

9 24 12 18

2 4 6 21 15

Figure 5: The zero-divisor graphs and zero-divisor lattices of Z8 and Z27

i = j, and if ann(ri) ⊆ ann(rj) for any i = j, then ann(si) ⊆ ann(sj). Thus, the mapping of edges φ : E(Λ(R)) → E(Λ(S)) which sends the edger ¯i −→ r¯j in Λ(R) to the edges ¯i −→ s¯j in Λ(S) is a well-deﬁned bijection. Thus, Λ(R) ∼= Λ(S).

We believe the converse to this theorem is true, because V (Γc(R)) = V (Λ(R)), which remedies the reason why the converse for both Theorems 3.1 and 4.2 are false. However, this remains an open question.

5 Connectivity of Λ(R)

By Theorem 3(4) on page 752 in [10], any ﬁnite commutative ring R with identity can be written as R ∼= L1 × L2 ×× Ln × F1 × F2 ×× Fm, where each Li is local and each Fj is a ﬁeld. We will use this fact in the upcoming results.

Theorem 5.1. Let M1,M2,...,Mn be ideals of a commutative ring R that is not a ﬁeld. Then M1,M2,...,Mn are the maximal ideals in R if and only if Λ(R) has n roots.

Proof. (⇒) Let M1,M2,...,Mn be the maximal ideals of R. By Corollary 2.6 and Lemma 2.8, Mi = ann(mi) for 1 ≤ i ≤ n. Since for any 1 ≤ i ≤ n, we have ann(mi) ann(mk) for all 1 ≤ k ≤ n, then mi is a root in Λ(R). Thus, Λ(R) has n roots. If there exists another root mr where r∈ / {1, 2,...,n}, then ann(mr) = Mr would be a maximal ideal in R. However, M1,M2,...,Mn are the only maximal ideals in R. (⇐) Let Λ(R) have n roots, namely m1,m2,...,mn. By Corollary 2.6, ann(x) is a maximal ideal for some x ∈ R. Obviously, | ann(x)|≥ 2 since R is not a ﬁeld, so x ∈ V (Λ(R)). Also, ann(x) is not properly contained in ann(r) for all r ∈ V (Λ(R)). So, ann(x) is a root of Page66 RHITUndergrad.Math.J.,Vol. 12, no. 1

Λ(R). Therefore, ann(x) = ann(mi) for some 1 ≤ i ≤ n. Hence, R has at most n maximal ideals. By definition of a root, ann(mi) ann(r) for any 1 ≤ i ≤ n and every r ∈ V (Λ(R)). Thus ann(mi) = Mi is a maximal ideal of R for every 1 ≤ i ≤ n. Since all maximal ideals are annihilator ideals, we must have each ann(mi) is maximal. Thus, M1,M2,...,Mn are maximal in R. The next four lemmas will help us prove the connectedness of the zero-divisor lattice. For each of them, recall from Corollary 2.6 that the maximal ideal of a finite ring can be written as the annihilator of a single element. Remark 5.2. Since all rings considered here are finite, we cannot have an infinitely ascend- ing chain of ideals. Hence, if ann(x) ann(y), it is either the case that y → x, or there exist z1,z2,...,zn such that y → z1 → z2 →→ zn → x. Thus, to show x and y are connected in a lattice (treated as an undirected graph), it suffices to show, without loss of generality, that ann(x) ann(y). This fact will be used in the following results. Lemma 5.3. If R is a finite local ring, then Λ(R) is connected. Proof. Let M be the maximal ideal of R. By Corollary 2.6 we know that M = ann(x), where x ∈ R, so we know thatx ¯ is a vertex of Λ(R). Also, for any other vertexy ¯ in the lattice, ann(y) ⊂ ann(x) since ann(x)= M. Thus,y ¯ < x¯.

Lemma 5.4. If R ∼= L1 × L2, where L1, L2 are ﬁnite local rings, then Λ(R) is connected.

Proof. By Lemma 2.7 and 2.8, we can write the maximal ideals of R as M1 × L2 = ann((m1, 0)) and L1 × M2 = ann((0,m2)), where Mi is the maximal ideal of Li. Since these are the only maximal ideals in R, all other ideals (and therefore all other annihilator sets) are subsets of either M1 × L2 or L1 × M2. This means that the vertices (m1, 0) and (0,m2) are roots of Λ(R). To show that Λ(R) is connected, we need to show that there exists a vertex x with x < (m1, 0) and x < (0,m2). Notice ann((m1, 0)) = {(l1,y) | m1l1 = 0 and y ∈ L2} and ann((0,m2)) = {(x,l2) | m2l2 = 0 and x ∈ L1}. Further notice that ann((m1, 1)) = {(l1, 0) | m1l1 = 0}, that ann((m1, 1)) ann((m1, 0)), and that ann((m1, 1)) ann((0,m2)). Thus, (m1, 1) < (m1, 0) and (m1, 1) < (0,m2), and Λ(R) is connected. Lemma 5.5. If R ∼= L × F , where L is a finite local ring and F is a finite field, then Λ(R) is connected.

Proof. By Lemma 2.7 and 2.8, we can write the maximal ideals of R as M1 ×F = ann((l, 0)), where M1 = ann(l) is the unique maximal ideal of L, and L × {0} = ann((0, 1)). Since these are the only maximal ideals in R, all other ideals (and therefore all other annihilator sets) are subsets of either M1 × F or L × 0. Thus, (l, 0) and (0, 1) are the roots of Λ(R). Notice that ann((l, 0)) = {(k,y) | kl = 0 and y ∈ F } and ann((0, 1)) = {(x, 0) | x ∈ L}. In order to show that Λ(R) is connected, we need to show the annihilator of some element is a proper subset of the annihilator sets of (l, 0) and (0, 1). Consider ann((l, 1)) = {(k, 0) | kl = 0}. Obviously, ann((l, 1)) ann((l, 0)) and ann((l, 1)) ann((0, 1)); therefore, (l, 1) < (l, 0) and (l, 1) < (0, 1), and Λ(R) is connected. RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 67

Lemma 5.6. If R ∼= R1 × R2 × R3, where R1, R2, R3 are ﬁnite commutative rings, then Λ(R) is connected.

Proof. By Lemma 2.7 and 2.8, we can write the maximal ideals of R as M1 × R2 × R3 = ann((m1, 0, 0)), R1 × M2 × R3 = ann((0,m2, 0)), and R1 × R2 × M3 = ann((0, 0,m3)) where Mi = ann(mi) for i = 1, 2, 3. Notice that ann((m1, 0, 0)) = {(l1,y,z) | m1l1 = 0, y ∈ R2, z ∈ R3}, ann((0,m2, 0)) = {(x,l2,z) | m2l2 = 0, x ∈ R1, z ∈ R3}, and ann((0, 0,m3)) = {(x,y,l3) | m3l3 = 0, x ∈ R1, y ∈ R2}. To show that Λ(R) is connected, we will find a vertex whose annihilator set is a subset of each of the maximal ideals. Con- sider ann((m1, 1, 1)) = {(l1, 0, 0) | m1l1 = 0}. Notice that ann((m1, 1, 1)) ann((m1, 0, 0)), ann((m1, 1, 1)) ann((0,m2, 0)), and ann((m1, 1, 1)) ann((0, 0,m3)). Thus, (m1, 1, 1) < (m1, 0, 0), (0,m2, 0), (0, 0,m3), and Λ(R) is connected. If R1 has more than one maximal ideal. We can simply consider the vertex (1,m2, 1) ′ since ann((1,m2, 1)) ann((m1, 0, 0)) and ann((1,m2, 1)) ann((m1, 0, 0)). Thus, Λ(R) is connected. Now we prove that the zero-divisor lattice is connected in all but one specific case. Theorem 5.7. Let R be a finite commutative ring. Then Λ(R) is connected if and only if R ≇ F1 × F2 for fields F1 and F2.

Proof. (⇐) Lemmas 5.3, 5.4, 5.5, and 5.6 show that if R ≇ F1 × F2, then Λ(R) is connected. (⇒) It suﬃces to show that if R ∼= F1 × F2, then Λ(R) is disconnected. Let R ∼= F1 × F2. Then there exists two maximal ideals, namely F1×0and0×F2. Again, each is the annihilator of a single element, call them (0, 1) and (1, 0), respectively. Since these are the only maximal ideals in R, all other ideals (and therefore all other annihilator sets) are subsets of either F1×0 or 0×F2. To show that Λ(R) is not connected, we need to show that there is no vertex whose annihilator set is a subset of both maximal ideals. Notice ann((0, 1)) = {(f, 0) | f ∈ F1} and ann((1, 0)) = {(0,f) | f ∈ F2}. Further notice that ann((0, 1)) ∩ ann((1, 0)) = {(0, 0)}, and the only elements in the ring whose annihilator set is the zero element are units. Since we do not allow units in the zero-divisor lattice, there is no vertex whose annihilator set is a subset of both maximal ideals; thus, Λ(R) is disconnected.

Corollary 5.8. Let Zpq be a ring with integers p and q. Then Λ(Zpq) is disconnected if and only if p and q are distinct primes.

Example 5.9. In Z6 ×Z9, the cut-sets are (2, 0), (3, 0), and (0, 3) in Γc(Z6 ×Z9). Notice that these vertices are the roots of Λ(Z6 × Z9). So, the cut-sets of the graph are easily identiﬁable by looking at the lattice. Figure 6 shows these graphs.

6 Conclusion

The hope of studying zero-divisor lattices is that it will be another way to identify the cut- sets of the zero-divisor graph. For example, in some cases of direct products of rings, we can Page68 RHITUndergrad.Math.J.,Vol. 12, no. 1

83, 1< 80, 1< 81, 0< 83, 0< 80, 3< 82, 0< 82, 0<

83, 3< 80, 3< 83, 3< 81, 0< 80, 1< 82, 3<

83, 0< 82, 3< 81, 3<

82, 1< 83, 1< 81, 3< 82, 1<

(a) Γc(Z6 × Z9) (b) Λ(Z6 × Z9)

Figure 6: The compressed zero-divisor graph and zero-divisor lattice of Z6 × Z9 quickly identify the cut-sets of the compressed zero-divisor graph by looking at the roots of the zero-divisor lattice, as is the case in Example 5.9. For future research we will attempt to create an algorithm that will identify the cut-sets of the zero-divisor graph by using the zero-divisor lattice. The motiviation behind the zero-divisor lattice arose from trying to solve the problem where in a finite local ring, a cut-set union with 0, does not always form an ideal. Notice that in the compressed zero-divisor graph of Z40, Figure 7, there are two cut-sets, {20} and {8}. Notice further that these are the roots of the zero-divisor lattice of Z40, shown in Figure 8. This also occurs with Z30, as you can check in Figures 1(b) and 2. Because the cut-sets seem to be easily identifiable in the zero-divisor lattice, further research into zero-divisor lattices may shed light upon the problem with the cut-sets in finite local rings.

4 10

2 20 8 5

Figure 7: Γc(Z40)

References

[1] Akbari, S., Maimani, H. R., Yassemi, S., When a zero-divisor graph is planar or a complete r-partite graph. J. Algebra 270(1) (2003), 169-180. [2] Anderson, D. F., Axtell, M., Stickles, J., Zero-divisor graphs in commutative rings. In Commutative algebra: Noetherian and non-Noetherian perspectives. (2010), Springer- Verlag. RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 69

20 8

10 4

5 2

Figure 8: Λ(Z40)

[3] Anderson, David F. and Livingston, Philip S., The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434-447.

[4] Anderson, D. D. and Naseer, M., Beck’s coloring of a commutative ring. J. Algebra 159 (1993), 500-514.

[5] Axtell, M., Baeth, N., Redmond, S., Stickles, J., Cut sets of commutative rings. preprint.

[6] Axtell, M., Stickles, J., Trampbachls, W., Zero-divisor ideals and realizable zero-divisor graphs, Involve 2(1) (2009), 17-27.

[7] Beck, I., Coloring of commutative rings. J. Algebra 116 (1988), 208-226.

[8] Chartrand, G., Introductory Graph Theory. Dover Publications, Inc. New York: 1977.

[9] Cot´e, B., Ewing, C., Huhn, M., Plaut, C. M., Weber, D., Cut-sets in zero-divisor graphs of ﬁnite commutative rings. Comm. Alg, to appear.

[10] Dummit, David S., and Foote, Richard M., Abstract Algebra, 3rd Ed. John Wiley and Sons, Inc. Hoboken, New Jersey: 2004.

[11] Kaplansky, Irving, Commutative Rings. Polygonal Publishing House. Washington, New Jersey: 1970.

[12] Mulay, S.B., Cycles and symmetries of zero-divisors, Comm. Alg. 30 (2002), no. 7, 3533-3558.

[13] Spiroﬀ, S. and Wickham, C., A zero divisor graph determined by equivalence classes of zero divisors. Comm. Alg. to appear. Page70 RHITUndergrad.Math.J.,Vol. 12, no. 1

[14] Weber, D., Mathematica Notebooks for Zero-Divisor Graphs. http://sites.google.com/site/zdgraphsformathematica/

[15] Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008).